# Influence of a time series on another one/clearing up trend or seasonality

I've got data as follows

| Date         |    Turnover    |    Day of week    |    Max Temp |
-------------------------------------------------------------------
| 2013-07-08  |    133          |    Monday         |    22       |
| 2013-07-09  |    150          |    Tuesday        |    23       |
| 2013-07-10  |    161          |    Wednesday      |    22       |
| 2013-07-11  |    127          |    Thursday       |    18       |


Now what I need to do is to find out how much influence the Max Temperature has on the turnover. The effects of the time series like trends and day of week (periodicity) should be cleared up before. So that in the end I want to be able to tell that the max temperature influences the turnover either positive or negative when it is between 10-15, 15-20, etc.

What I already did with R is to do a time series decomposition. My next plan was to then take the irregular "noise" component from the time series (which as I understand is without any periodicity or trend) and see how the temperature influences the turnover. But here I don't know which statistical method is best suited for this. I need some method to cluster the data maybe.

Any hints?

A Vector Autoregressive (VAR) analysis might be suitable for this problem. If you are familiar with univariate autoregressive models, this is pretty much straight forward. If your dependent variable is the Turnover $T$, the VAR model regress on the lagged values of the Turnover and the Max Temp $M$. Your model would look like, $T_t=\beta_0+\beta_1 T_{t-1}+\gamma_1 M_{t-1}+\cdots+\beta_p T_{t-p}+\gamma_p M_{t-p}+\epsilon_t$ From here you can interpret the coefficients of the lagged values of both your variables like OLS variables. For the termperature intervals you can add dummy variables, for example, turnover between $10-15$ the dummy is $1$ otherwise $0$, etc.
Or you could just fit a linear model, $T=\beta_0+\beta_1M+\epsilon$. Either way, check the time series for unit root, otherwise you might end up with a spurious regression.